> If you shaped the bags into uniform-sized polyhedrons and fit them together seamlessly, like bricks, to form this growing tetrahedron, then you would be showing what the cube people show. The cube people show smaller bricks making the bigger cube. Here, we have an octahedron and tetrahedron that together fill space.> > Another demo would be to take a glass or clear plastic tetrahedron and bury the tip in the sand so it stays upside down. Pour liquid into it using a unit tetrahedron measuring cup. Mark the positions 1, 8, 27, 64... and notice they are equally spaced up the side. > > Unlike a cube, when you slice a regular tetrahedron parallel to any side, you still have a regular tetrahedron. Nice property. Slice a cube that way and you have a cube no longer.

No, there is a lot more to it than that Kirby. That is probably why just stacking stuff never caught on with teaching mathematics. Like I said, you are confusing intuition[1] with rhetoric[2]. What point is there in displaying examples that the student couldn't possibly understand the underlying reasons for? The point was to teach about cubing, not to perform a magic show. I think I stick with the old fashioned approach of teaching cubing first, and several other things, and then later we can really dive into the analytic geometry behind tetrahedrons.

Bob Hansen

[1] intuition: to sense that something is true[2] rhetoric: to say that something is true